Question 1209701
.
Find a monic quartic polynomial f(x) with rational coefficients whose roots 
include x = 2 - i \sqrt[3]{3}$. Give your answer in expanded form.
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The solution in the post by @CPhill is INCORRECT.


<pre>
{{{highlight(highlight(First))}}}, I am not sure whether he correctly interprets x = 2 - i \sqrt[3]{3}$ in the condition 

as 2 - i∛3.   This fragment of the condition is, actually, UNREADABLE in this forum format, 

so I ask you to write your formulas in plain text format, to avoid misreading.



{{{highlight(highlight(Second))}}}, if to take this interpretation of @CPhill, then in the process of the solution he makes 
a fatal error.


Indeed, he writes in his post

    **3. Eliminate the Cube Root:**

    To get rational coefficients, we need to eliminate the ∛9 term. Let's call the quadratic factor we just found 
         g(x) = x² - 4x + 4 + ∛9.   To   eliminate the cube root, we'll work with the equation  ∛9 = -x² + 4x - 4, and cube both sides:
    9 = (-x² + 4x - 4)³


It is totally wrong.  From  g(x) = x² - 4x + 4 + ∛9,  we only can express  ∛9 = g(x) -  (x² - 4x + 4),

but this way will lead us to NOWHERE.
</pre>

So, &nbsp;it is the point, &nbsp;where @CPhill makes this fundamental error, &nbsp;which ruines all his solution to dust.


My condolences. &nbsp;Ignore the solution by &nbsp;@CPhill, &nbsp;since it is &nbsp;IRRELEVANT &nbsp;and leads you to &nbsp;NOWHERE.